Modular inequalities for the Calderón operator

Abstract

If P, Q:(0, ∞)→ are increasing functions and T is the Calderon operator defined on positive or decreasing functions, then optimal modular inequalities ∫ P(Tf)≤ C∫ Q(f) are proved. If P=Q, the condition on P is both necessary and sufficient for the modular inequality. In addition, we establish general interpolation theorems for modular spaces.